We introduce a notion of separativity for positively ordered monoids (POMs), similar in definition to the notion of separativity for commutative semigroups but which has a simple categorical equivalent, weaker that injectivity, the transfer property. We show that existence in a separative extension of the ground POM of a solution of a given linear system is equivalent to the satisfaction by the ground POM of a certain set of equations and inequalities, the resolvent. We deduce in particular a characterization of the POMs that are injective relatively to the class of embeddings of countable POMs; those include in particular divisible weak cardinal algebras. We also deduce that finitely additive positive non-standard measures invariant relatively to a given exponentially bounded group separate equidecomposability types modulo this group.